Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

PRELIMINARY CALCULUS


a b c

f(x)

x

Figure 2.5 The graph of a functionf(x), showing that iff(a)=f(c)thenat
one point at least betweenx=aandx=cthe graph has zero gradient.

a b c

C


f(a) A

f(x)

x

f(c)

Figure 2.6 The graph of a functionf(x); at some pointx=bit has the same
gradient as the lineAC.

Mean value theorem

The mean value theorem (figure 2.6) states that if a functionf(x) is continuous


in the rangea≤x≤cand differentiable in the rangea<x<cthen


f′(b)=

f(c)−f(a)
c−a

, (2.20)

for at least one valuebwherea<b<c. Thus the mean value theorem states


that for a well-behaved function the gradient of the line joining two points on the


curve is equal to the slope of the tangent to the curve for at least one intervening


point.


The proof of the mean value theorem is found by examination of figure 2.6, as

follows. The equation of the lineACis


g(x)=f(a)+(x−a)

f(c)−f(a)
c−a

,
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